the production theory approach to import demand...

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THE PRODUCTION THEORY APPROACH TO IMPORT DEMAND ANALYSIS: A COMPARISON OF THE ROTTERDAM MODEL AND THE DIFFERENTIAL PRODUCTION APPROACH

By

Andrew A. Washington and Richard L. Kilmer

JOURNAL REPRINT SERIES

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INTERNATIONAL AGRICULTURAL TRADE AND POLICY CENTER

MISSION AND SCOPE: The International Agricultural Trade and Policy Center (IATPC) was established in 1990 in the Food and Resource Economics Department (FRED) of the Institute of Food and Agricultural Sciences (IFAS) at the University of Florida. Its mission is to provide information, education, and research directed to immediate and long-term enhancement and sustainability of international trade and natural resource use. Its scope includes not only trade and related policy issues, but also agricultural, rural, resource, environmental, food, state, national and international policies, regulations, and issues that influence trade and development. OBJECTIVES: The Center’s objectives are to:

• Serve as a university-wide focal point and resource base for research on international agricultural trade and trade policy issues

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federal agencies, and policymakers in the examination and discussion of agricultural trade policy questions

• Provide support to initiatives that enable a better understanding of trade and policy issues that impact the competitiveness of Florida and southeastern agriculture specialty crops and livestock in the U.S. and international markets

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The Production Theory Approach to Import Demand Analysis: A Comparison of the

Rotterdam Model and the Differential Production Approach

by

Andrew A. Washington (Assistant Professor, Department of Economics, Southern University)

and

Richard L. Kilmer (Professor, Food and Resource Economics Department, University of Florida and a member of the International Agricultural Trade and Policy Center (IATPC) at the University of Florida)

Citation

This article was published in the Journal of Agricultural and Applied Economics, 34(3)(December 2002), Washington, Andrew A. and Richard L. Kilmer, "The Production Theory Approach to Import Demand Analysis: A Comparison of the Rotterdam Model and the Differential Production Approach”, pages 431-443, Copyright 2002, and is posted with the permission of the Southern Agricultural Economics Association, http://www.agecon.uga.edu/~jaae/. Copies of the article can be downloaded and printed only for the reader’s personal research and study.

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Abstract

Results indicate that when comparing the unconditional derived demand elasticities to the

unconditional consumer demand elasticities, significant differences emerge due to the

differences in the first stage estimation procedure between the differential production approach

and the Rotterdam model. In comparing the consumer demand price/cross price elasticities to the

derived demand price/cross price elasticities it is clear that use of the Rotterdam model when a

production approach should be used can lead to overestimation, underestimation and incorrect

signs in deriving unconditional price effects.

Key Words: Dairy, demand, imports, international, production, Rotterdam, trade

JEL Classifications: D12, D24, F10, F14, Q17

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The Production Theory Approach to Import Demand Analysis: A Comparison of the

Rotterdam Model and the Differential Production Approach

The Rotterdam model application to import demand has been accomplished by a number of

studies (Lee, Seale, and Jierwiriyapant; Seale, Sparks, and Buxton; Zhang, Fletcher, and Carley).

In past studies, imports are considered to be final goods that enter directly into the consumer's

utility function and the resulting demand equations for imports are derived from utility

maximization theory. However, given the nature of international trade, where traded goods are

either used in other production processes or go through a number of domestic channels before

reaching the consumer, it is more appropriate to view imported goods as intermediate products

than as final consumption goods even if no transformation takes place (Davis and Jensen). The

primary objective of this paper is to compare and contrast the use of the differential production

approach with the Rotterdam model. Both approaches are applied to Japan's derived demand for

imported whey differentiated by source country of production. Unconditional elasticities from

both approaches are then compared.

The application of production theory to international trade is by no means a new concept.

Past research that used a production theory approach to international trade include Burgess

(1974a) and (1974b), Kohli (1978) and (1991), Diewert and Morrison, and Truett and Truett.

Each of these studies acknowledged that most goods entering into international trade require

further processing before final demand delivery. They further acknowledged that even when a

traded product is not physically altered, activities such as handling, insurance, transportation,

storing, repackaging, and retailing still occur. This results in a significant amount of domestic

value added when the final product reaches the consumer. Therefore it is more appropriate to

view imported products as inputs rather than as final goods even if goods are not transformed.

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Davis and Jensen (pp. 410-12) meticulously discuss the advantages of the production

theory approach over the utility approach to import demand estimation. Their first point is that

most imported agricultural commodities are inputs and not final goods. Second, specifying the

first stage aggregates is more intuitive when using the production theory approach. Third, it is

easier and more intuitive to estimate unconditional elasticities using production theory. Their last

point is that the estimated parameters using production theory will be structural parameters.1

Kohli (1991) notes that viewing imports as intermediate goods not only has its merits in

correctness, but it also leads to substantial simplifications theoretically. One simplification is that

the demand for imports can be derived from production theory and there is no need to model

final demand. Second this approach allows for the avoidance of the difficulties that arise when

we aggregate over individual consumers. To expound on this point, data is typically reported in

aggregate terms. Therefore, if we are estimating demand, we are estimating aggregate demand,

and if we are estimating derived demand it is aggregate or industry derived demand. The

differences between aggregate demand and aggregate derived demand is that one is an

aggregation over consumers and the latter is an aggregation over firms. When we consider

optimizing behavior by both consumers and firms, do the properties derived from consumer and

producer-maximizing behavior hold in the aggregate? Mas-Colell, Whinston, and Green indicate

that when consumer preferences and wealth effects are identical across consumers, the aggregate

demand function satisfies all of the properties of an individual demand function.2 However, if

there is the slightest difference in preferences and if these differences are independent across

consumers (as one would expect), the property of symmetry, which is a common property tested

in most empirical demand studies, will almost certainly not hold.3

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When we aggregate across firms, there are no such conditions required for the properties

of optimal firm behavior to hold in aggregation. This is because the aggregate profit obtained

when each production unit maximizing profit separately, taking prices as given, is the same as

that which would be obtained if they were to coordinate their actions in a joint profit maximizing

decision (Mas-Colell, Whinston, and Green).4 This result implies that the profit maximizing

output arrived at if all firms coordinated their actions is the same as the sum of the individual

output of each profit-maximizing firm. It further implies that the total cost of production for the

coordinated output is the same as the sum of total cost for each individual firm if firms are price

takers in the input market (Mas-Colell, Whinston, and Green). Therefore, if we estimate input

demand functions and output supply functions using aggregate data, the properties of the demand

and supply functions for each individual firm will theoretically hold in aggregation.5

Overview of Theory

The differential approach to the theory of the firm is comparable to the differential

approach to consumer theory proposed by Barten (1964) and Theil (1965). The empirical

application of the differential approach to consumer demand resulted in the Rotterdam model,

which has been used extensively in demand studies and to a lesser extent in import demand

studies. The majority of import demand studies that used the Rotterdam model assumed that

imported goods entered directly into the consumer's utility function and strong assumptions were

made about how consumers view imported and domestic goods and how they grouped

commodities. Furthermore, it was often assumed that these commodity groups were to some

degree independent in terms of the consumer's utility function (For example, see Lee, Seale, and

Jierwiriyapant; Seale, Sparks, and Buxton; and Zhang, Fletcher, and Carley). In these studies, the

intermediate nature of imports was not considered.

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The Rotterdam Model

The estimation of import demand using the Rotterdam model is accomplished in two

stages. First consumers allocate total expenditures between product groups (first stage) and

second, consumers allocate total group expenditures among goods within the product group

(second stage).6 It is also assumed that product groups are blockwise dependent, that is the

utility interaction among goods are a matter of the groups and not the individual goods.

The first stage of the consumer budgeting process results in a system of composite

demand equations where each equation is expressed as

(1) )(log)(log)(log '

1hgh

G

hggg PdQdQdW Π+Θ= ∑

=

,

where )(log gQd and )(log 'hPd are the group Divisia volume and Frisch price indexes

respectively; gW , gΘ , and gΠ are the budget share, marginal share, and absolute price

coefficient respectively; )(log Qd is the percentage change in real income (Theil, 1980, p. 101).

Equation (1) states that the composite demand for the product group depends on real income and

the Frisch price indexes for each group. The size of the system represented by equation 1 is equal

to the total number of groups specified in the consumer's utility function. When estimating

import demand, the total number of equations in the system can be as large as the total number of

goods imported which makes estimating equation (1) problematic.

The demand for individual goods within a group conditional on total group expenditures

(second stage) results in a system of demand equations where each equation is expressed as

(2) ∑=

+=n

jjijgiii pdQdqdw

1)(log)(log)(log πθ ,

where wi represents the share of group expenditures allocated to good i and iθ is the conditional

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marginal share; iq and jp are the quantities and prices, respectively; ijπ 's are the conditional

Slutsky price coefficients; and n is the number of goods within the product group (Theil, p. 103).

Dividing equation (1) by gW and substituting into equation (2) yields the unconditional

demand equation

(3) ∑∑==

+Π

+Θ

=n

jjijh

g

ghG

hg

giii pdPd

WQd

Wqdw

1

'

1)(log)](log)(log[)(log πθ .

From equation (3) we get the unconditional income elasticity

(4) g

g

i

iii WwQd

qd Θ==

θη

)(log)(log

,

which is the product of the conditional expenditure elasticity ii wθ and the expenditure

elasticity for the group gg WΘ . We also get the unconditional price elasticity

(5) i

ij

g

gh

i

i

j

iij wWwpd

qd πθη +

Π==

)(log)(log

,

where ggh WΠ is the own-price elasticity for the group and iij wπ is the conditional price

elasticity for the ith good.

The Differential Production Approach

Using the methodology of Laitinen and Theil, Laitinen, and Theil (1980), the differential

production model will also be used to estimate the import demand. The differential production

model is derived from the differential approach to the theory of the firm where firms maximize

profit in a two-stage procedure. In the first stage, firms determine the profit maximizing level of

output to produce and in the second stage firms minimize the cost of producing the profit

maximizing level of output. According to Laitinen and Theil, and Davis and Jensen, this

procedure is consistent with a one-step or direct profit maximization procedure. In the first stage

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the output supply equation is obtained and the conditional factor demand system is obtained in

the second stage. Using the results of both stages, a system of unconditional derived demand

equations is derived.

In the first stage a competitive firm seeks to identify the profit-maximizing level of

output by equating marginal cost with marginal revenue. This procedure yields the differential

output supply equation

(6) ∑=

+=N

jjj wdpdQd

1

** )(log)(log)(log πϕ ,

where Q*, p* and wi represent the output, output price and the price of inputs respectively; ϕ and

π are the price elasticity of supply and the elasticity of supply with respect to input prices

respectively. N is the total number of inputs used in production.

In the second stage, the differential factor demand model is derived, which will be used

to estimate the system of source specific derived demand equations. This model is specified as

(7) ∑=

+=n

jjijiii wdXdxdf

1

** )(log)(log)(log πθ ,

where if is the factor share of imported good x from source country i in total input cost; xi and wi

represent the quantity and price of inputs which include the price of each imported good from

source country i; ∑=

=n

itit xdfXd

1)(log)(log where )(log Xd is the Divisia volume input

index;θ i* is the mean share of the ith input in the marginal cost of the firm;π ij

* is the conditional

price coefficient between the ith and jth importing sources or inputs; n is the number of inputs in

the system, n ∈ N.7

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The differential factor demand model requires that the following parameter restrictions be

met in order for the model to conform to theoretical considerations:j

ij∑ =π * 0 (homogeneity),

and **jiij ππ = (symmetry). The second stage procedure results in the conditional own price/cross

price elasticity

(8) i

ij

j

icxw fwd

xd *

)(log)(log π

ε == ,

and the conditional Divisia volume input elasticity,

(9) i

iixX fXd

xd *

)(log)(log θ

ε == .

Using the relationship between the Divisia volume input index and output,

)(log)(log *QdXd γ= 8, equation (6) can be substituted into equation (7) to yield the

unconditional derived demand system

(10) ∑∑==

++=n

jjijj

n

jjiii wdwdpdxdf

1

*

1

** )(log)](log)(log[)(log ππϕγθ .

Dividing through equation (10) by if and using equations (8) and (9) we get the unconditional

derived demand elasticities. The unconditional elasticity of input demand with respect to output

price is

(11) ϕγεε xXi

xp pdxd

==)(log)(log

*,

and the unconditional own price/cross price elasticity of input demand is

(12) cxwjxX

j

ixw wd

xdεπεγε +==

)(log)(log

.

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Lastly we get the unconditional elasticity of derived demand with respect to the price of an input

contained in N but not in n

(13) jxXj

ixw wd

xdπγεε ==

)(log)(log

.

Inputs contained in N but not in n include labor and other inputs that are not part of the imported

whey group.

The second stage procedures in the consumer and production approaches yield

empirically identical demand systems, equation (2) and equation (7), resulting in identical

conditional elasticities. Davis and Jensen note that this similarity explains the empirical success

of consumer based conditional demand systems even though they may be conceptually flawed.

However given the differences in the first stage, equation (1) and equation (6), unconditional

elasticities differ between the two approaches. Also, the production approach results in the

unconditional elasticity of derived demand with respect to output price whereas the Rotterdam

model results in the unconditional income elasticity. This suggests that the use of the Rotterdam

model, when a production approach is more appropriate, not only leads to biased unconditional

own price/cross price elasticity estimates but also leads to the reporting of unconditional income

elasticities when the concern should the unconditional elasticity of derived demand with respect to

output price.

Application to the Derived Demand for Imported Whey in Japan

This study assesses the competitiveness of whey imports into Japan from the U.S. compared to

whey imported from other countries such as the EU, Australia, and New Zealand. Following

Armington, similar imported dairy products such as EU whey and US whey are both individual

goods that are part of the product group whey, but different based on their source country of

production. There are a number of reasons why similar products are viewed as different based on

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their source country of origin. Dairy products from different sources may actually be physically

different. Physical differences include quality, protein, fat content, and taste. There may also be

perceived differences, such as a country’s reputation for a quality product, trade history,

reliability and consistency, and political issues tied to trade (Zhou and Novakovic). The crux of

this assumption is that within an importing country, a particular dairy product imported from a

given source is considered a substitute for that same product from another source. However,

because of the physical and perceived differences attributed to the product due to its origin, these

products are imperfect substitutes.

In this paper it is assumed that dairy products are imported through firms that exclusively

import. Although, there are firms within Japan that import whey as well as transform whey into

other products, it is assume that there is a separate entity within the firm that deals primarily with

the procurement of imported dairy products. Also, dairy imports through this type of firm make

up a smaller percentage of imports in Japan. In addition to providing imported products to other

firms, these firms also provide the services that are associated with importing. These services

include, search and acquisition, transportation, logistics, and storing. A major characteristic of

this firm type is that it deals primarily in imported goods. This suggests that the procurement of

imported goods by firms is a unique process separate from the procurement of similar products

produced domestically. Even if the firm is a subsidiary or branch of a larger firm that purchases

domestic and foreign produced inputs, it is not unlikely that the subsidiary that is responsible for

imported inputs deals primarily in this activity. This is because the acquisition of foreign

produced goods is more involved than purchasing domestically produced goods.

If we assume a production function for these firms, then the output of these firms is the imported

goods that are sold to other firms and the inputs are the imported goods from the various

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exporting countries. If we minimize cost subject to this production function, the system of input

demand equations resulting from the optimization procedure will be a system of import demand

equations. If we assume product differentiation across source countries, then each import

demand equation represents the demand for a product from a particular source.

In the first-stage, the importing firm seeks to maximize profit by equating marginal cost

with marginal revenue. This procedure yields the differential output supply equation (expressed

in finite log changes)

(14) ∑=

++=N

jitjtj

*t

*t wpQ

1

ε∆π∆ϕ∆ ,

where )/log( 1−=∆ ttt QQQ , )/log( 1−=∆ ttt ppp and )/log( 1−=∆ ititit www , where q, p and wi's

represent the output, output price and input prices; ϕ and π are the parameters to be estimated

which are also the own-price elasticity of supply and the elasticity of supply with respect to input

prices respectively; ε it is the disturbance term. Q* represents Japan's total imports of whey that

is to be supplied, p is the price at which firms in Japan sell whey, and the wi’s are the prices paid

for whey imports from each of the exporting countries, the price of labor (wages), and the price

of other inputs used. N is the total number of inputs used in production.

In the second stage, the differential factor demand model is derived, which is used to

estimate the system of derived demand equations where each equation is the derived demand for

imported whey from a particular source. This model is specified as follows (expressed in finite

log changes)

(15) ∑=

+∆+∆=∆n

jitjtijtiitit wXxf

1

**_

επθ ,

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where 2/)( 1

_

−+= ititit fff ; )/log( 1−=∆ ititit xxx and )/log( 1−=∆ ititit www , where xi and wi

represent the quantity and price of imported whey from source country i;

∑=

∆=∆n

iititt xfX

1

_

where tX∆ is the finite version Divisia volume input index;θ i* and π ij

*

parameters to be estimated; n is the number of inputs in the system; ε it is the disturbance term.

In addition to the imports from each individual source country, labor and other inputs are

used in the production process. The labor demand and demand for other inputs are expressed in

general terms as

(16) Labor = f (output, wages, input price index)

(17) Other Inputs = f (output, wages, input price index).

Equations (16) and (17) represent the system of derived demand equations for labor and other

inputs where these inputs are a function of the total amount to be supplied, wages, and an input

price index which represents the price of all inputs except labor and whey imports. Here we

assume that labor and other inputs are independent of the source specific whey imports. This is

to say that although labor and other inputs affect the total to be imported, these inputs do not

directly affect the amount imported from an individual source country.

Empirical Results

Using United Nations Commodity Trade Statistics, the derived demand for imported whey into

Japan was estimated. The exporting countries considered were the United States, European

Union, Oceania (aggregation of Australia and New Zealand), and rest of the world (ROW),

which is an aggregation of all other countries. The time period for the data set was 1976 to 1998.

During this period, the United States on average accounted for 35% of all whey exports to Japan,

while Oceania, EU, and ROW accounted for 17, 19 and 27%, respectively. All values and

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quantities where reported through Japanese customs. Values were on a cost, insurance, and

freight basis. According to FAO statistics, Japan primarily imports dry whey, which is used as

both cattle feed and an ingredient in infant formula. In the last decade, imports of dry whey have

accounted for 100% of all whey imports.

First stage estimation required the domestic wholesale price of whey in Japan. This price

series was not available. However, a proxy was used which was the per unit wholesale price of

all milk powders which is reported by the Statistic Bureau Management and Coordination

Agency for the Government of Japan. To account for the labor requirement in the importation of

whey, an index of Japan's hourly wages was included in the estimation (U.S. Department of

Labor). To account for other inputs, an industry input price index was also included

(Economagic.com).

Second-Stage Estimation and Conditional Elasticities

Table 1 presents the log-likelihood values, the likelihood ratio (LR) statistics, and the critical

value for the LR test for autocorrelation. A Likelihood ratio test indicated that first-order

autocorrelation could not be rejected at the .05 significance level; thus, all results presented have

the AR(1) error structure imposed.9

[Place Table 1 approximately here]

LR tests were also used to test if the data satisfied the economic properties of

homogeneity and symmetry. The results of these tests are summarized in Table 2. LR tests

indicate that the property of homogeneity could be rejected. However, Laitinen’s test for

homogeneity, which is a more precise test, indicated that homogeneity could not be rejected.

Given the homogeneity constraint, symmetry could not be rejected. The property of negative

semi-definiteness was verified by inspection of the eigen values of the price coefficient matrix.

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This property is validated when all of the eigen values are less than or equal to zero. All eigen

values were non-positive in the Japan-whey system.


Table 3 presents the conditional parameter estimates for the derived demand and

consumer demand for imports of whey into Japan. With the exception of the ROW, all own-price

coefficients are negative and all are significant by at least the .05 significance level. The

condition marginal factor share estimates indicate a positive relationship between the Divisia

volume index of all imports and the imports from the individual sources except for the ROW.10

In the consumer demand (Rotterdam) model the conditional marginal factor shares are

interpreted as the conditional marginal expenditure share. Cross-price parameter estimates

indicate that the U.S and Oceania whey imports, Oceania and EU imports, and EU and ROW

imports are substitutes.


Table 4 presents the conditional elasticties for the derived demand and consumer demand

of imported whey.11 The Divisia index elasticities for imports of whey into Japan are .914, 2.295,

2.336 and -.500 for the U.S., Oceania, EU and the ROW, respectively. These indicate that as the

Divisia volume index increases, imports from the US will increase proportionately while imports

from Oceania and the EU will increase by more than twice as much. In the consumer demand

model, these are interpreted as conditional expenditure elasticities. The own-price elasticities are

-1.031, -2.930, -1.574, and -.296 for the U.S., Oceania, EU, ROW, respectively. With the

exception of the ROW, all are significant at the .10 significance level. Conditional cross-price

elasticities of derived demand for whey in Japan indicate significant substitutional relationships

between whey imports from the exporting sources. The U.S./Oceania cross-price elasticity is

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1.003, while the Oceania/U.S. elasticity is 2.106, reflecting the higher value placed on U.S.

whey. The Oceania/EU and the EU/Oceania elasticities are .441 and .401, respectively,

indicating fairly equal substitutability between the two sources. EU whey imports are the only

imports that were substitutes for whey from the ROW.


First-Stage Estimation and Unconditional Elasticities

First-stage estimation required the estimation of equation (14), which is the output supply

equation. Results are presented in Table 5. The output price parameter estimate (1.2963) is

positive as expected and significant at the .01 significance level. This estimate is also the price

elasticity of supply, which indicates that the supply of whey in Japan is price elastic. Parameter

estimates for all import prices are insignificant. The parameter estimate for the price of labor and

the price of other inputs (-1.4888 and -3.3351, respectively) are negative and significant

indicating that wages and other input prices are inversely related to the output supplied, which is

to be expected. These are also the elasticity of output supply with respect to the price labor and

with respect to the input price index. These indicate that the supply of imported whey in Japan is

relatively sensitive to wages and other input prices. First-stage estimation in the differential

production model is possible and correct estimates could be used to derive unconditional derived

demand elasticities.


Unconditional elasticities for the Rotterdam model and the unconditional derived demand

elasticities are presented in Tables 6 and 7, respectively. To derive the unconditional income

elasticities for the consumer demand (Rotterdam) model (equation (4)), the income elasticity for

the product group whey was estimated to be one.12 For the unconditional own price/cross price

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elasticities (equation (5)), it is assumed that the price elasticity of the demand for the product

group is -.40 (Zhu, Cox, and Chavas). Unconditional derived demand elasticities were derived

using equations (11), (12), and (13).



In comparing the unconditional Rotterdam elasticity estimates in Table 6 to the

unconditional derived demand elasticities in Table 7, the biasness due to the inappropriate

application of consumer theory to import demand analysis becomes clear. First, the elasticity of

derived demand with respect to output prices, the elasticity of derived demand with respect

wages, and the elasticity of derived demand with respect to other input prices would not be

considered if the consumer demand model were applied. These derived demand elasticities

suggest that the derived demand for whey is highly responsive to these factors.

In addition to not reporting some of the derived demand elasticities, the Rotterdam model

leads to substantial differences in the unconditional own price/cross price elasticities. In the case

of the own price elasticities, the Oceania and EU elasticities derived using the Rotterdam model

are substantially larger in absolute terms than the derived demand elasticities. In the case of the

own price elasticity of demand for Oceania whey, the Rotterdam model overstates the own-price

effect by 1.6 percentage points.

Unconditional cross price elasticities differ between the approaches as well. Of the 12

unconditional cross-price elasticities, 11 are significant in the derived demand model while 8 are

significant when using the Rotterdam model. Five cross-price elasticities actually change signs

(U.S./EU, Oceania/ROW, EU/Oceania, EU/ROW, and EU/U.S). The largest difference occurred

with EU/Oceania elasticity, which was estimated to be -.534 in the Rotterdam model and 1.114

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in the derived demand model. Using the Rotterdam elasticities, one would assess that EU whey

and Oceania whey were complements while the derived demand model indicates a substitutional

relationship.

Summary and Conclusions

The primary objective of this paper was to compare and contrast the use of the differential

production approach with the Rotterdam model. Given the intuitive and conceptual appeal of a

production approach to import demand analysis instead of a consumer approach (the Rotterdam

model), this article investigates the empirical differences due to approach selection. When one

compares the conditional derived demand to the conditional consumer demand system, there is

no empirical difference. However, when comparing the unconditional derived demand

elasticities to the unconditional consumer demand elasticities, significant differences emerge.

This is due to the differences in the first stage estimation procedure between the two approaches.

In fact, first stage estimation using the Rotterdam model is often not accomplished due to

difficulty in defining product groups that make up the first stage. However, in this study, it was

shown that first stage estimation is possible with the production approach and lead to

unconditional elasticity estimates. One empirical difference is that with the production approach,

the derived demand elasticity with respect to output price, wages, and other input prices are

derived. This is not the case with the Rotterdam model. In comparing the consumer demand own

price/cross price elasticities to the derived demand own price/cross price elasticities, it is clear

that use of the Rotterdam model when a production approach should be used can lead to

overestimation, underestimation, incorrect signs, and erroneous insignificance when deriving the

unconditional price effects.

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Footnotes

1. For a more in-depth discussion of the conceptual and theoretical advantages of the

production approach see Davis and Jensen.

2. The properties of a system of demand equations for a utility maximizing consumer are

adding up, homogeneity, and the symmetry and negative semi-definiteness of the matrix of

price effects.

3. The property of negative semi-definiteness holds in aggregation under less strict conditions.

If each individual demand function satisfies the uncompensated law of demand, then the

aggregate demand function satisfies the week axiom of revealed preference, which implies a

negative semi-definite price effect matrix.

4. Prices are assumed as given even with coordination.

5. The properties of the input demand function are the same as the properties of the consumer

demand function. The properties of the supply function are that the matrix of price effects is

symmetric and positive semi-definite. The authors assumed that firms are still price takers

even with coordination. Production technology can vary over firms.

6. Given a common assumption that imports and domestics goods are independent, there is an

additional stage before the two mentioned where total expenditures are allocated between

imports and domestic goods (Seale, Sparks, and Burton).

7. The derivation of equations (6) and (7) are found in Laitinen and Theil.

8. γ is the elasticity of cost with respect to a proportionate output increase. According to

Laitinen (p.113), γ is also the ratio of revenue to cost. When calculating elasticities, the

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average of the geometric mean of γ for periods t and t-1 is used, where tγ = 1

1

−

−

tt

tt

CCRR is the

two period geometric mean and ∑ == T

t t1γγ is the average of tγ across all observations.

9. The AR(1) process is the same for all equations in the system.

10. Homogeneity and symmetry are imposed on the parameters. AR(1) is also imposed.

11. All elasticities are evaluated at the mean.

12. The income elasticity for the group whey was estimated using the Workings Model (Theil

and Clements, p. 14). The income elasticity for the group whey was equal to one.

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REFERENCES

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Table 1. Likelihood Ratio Test Results for Autocorrelation in the Derived Demand and Consumer Demand Models

Country/Product Model Log-likelihood

Value

LR* P[ 2)( jχ ≤LR*]=.95

Japan-Whey AR(1) 55.125

No-AR(1) 48.729 12.7927 3.84(1)a

a The number of restrictions are in parentheses.

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Table 2. Likelihood Ratio Test Results for Economic Constraints and Laitinen’s Test For Homogeneity in the Derived Demand and Consumer Demand Models

Country/Product Model Log-likelihood

Value

LR* P[ 2)( jχ ≤LR*]=.95

Japan Whey Unrestricted 55.541 Homogeneity 51.179 8.726 7.81(3)a

Symmetry 48.998 4.362 7.81(3) Laitinen’s Test W*b P[T2≤W*]=.95c

Japan Whey Homogeneity 9.217 11.186 a The number of restrictions are in parentheses. b W* is the Wald statistic for the homogeneity constraint. c T2 is the Hotelling’s T2 statistic.

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Table 3. Conditional Derived Demand (Consumer Demand) Parameter Estimates for Japan Imports of Whey

Price Coefficients, )(and*ijij ππ

Exporting Country

U.S. Oceaniaa EU ROWb

Marginal Factor Shares,

)(and*ii θθ

U.S. -.3653*** (.1254)c

.3556*** (.0686)

.1032 (.0739)

-.0935 (.0884)

.3239** (.1729)

Oceania -.4947*** (.0973)

.0744** (.0426)

.0647 (.0836)

.3874*** (.0948)

EU -.2926** (.0628)

.1150* (.0649)

.4341*** (.1166)

ROW -.0862 (.1286)

-.1454 (.1228)

System R2 = .79

a Australia and New Zealand aggregation. b ROW= rest of the world. c Asymptotic standard errors are in parentheses. *** Significance level = .01. ** Significance level = .05. * Significance level = .10.

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Table 4. Conditional Divisia and Price Elasticities of the Derived Demand and Consumer Demand for Imported Whey

Elasticities

Conditional Cross-Price Exporting Country

Divisia Index

Conditional Own-Price

U.S. Oceaniaa EU ROWb U.S. .914*

(.488)c -1.031*** ( .354)

1.003*** ( .193)

.291 (.209)

-.264 (.249)

Oceania 2.295*** (.562)

-2.930*** ( .577)

2.106*** (.406)

.441* (.252)

.383 (.495)

EU 2.336*** (.627)

-1.574*** ( .338)

.555 (.397)

.401* (.229)

.618* (.349)

ROW -.500 (.422)

-.296 ( .442)

-.321 (.303)

.222 (.287)

.395* (.222)

a Australia and New Zealand aggregation. b ROW = rest of the world. c Asymptotic standard errors are in parentheses. *** Significance level = .01. ** Significance level = .05. * Significance level = .10. Note: A Wald statistic was used which has a χ2 distribution.

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Table 5. Parameter Estimates for the Supply of Whey in Japan

Input Price Coefficients, πij Output Price Coefficient

U.S. Oceaniaa EU ROWb Wage Input Price

Index

-.0322 (.0974)c

.1638 (.1477)

.0001 (.0670)

.0575 (.1890)

-.4888*** (.4143)

-3.3351** (1.6403)

1.2963*** (.3709)

R2 = .57

a Australia and New Zealand aggregation. b ROW= rest of the world. c Asymptotic standard errors are in parentheses. *** Significance level = .01. ** Significance level = .05. * Significance level = .10.

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Table 6. Unconditional Elasticities of the Consumer Demand Model (Rotterdam Model) Elasticities

Cross-Price Exporting Country

Income Own-PriceU.S. Oceaniaa EU ROWb

U.S. .914* (.488)c

-1.396*** ( .195)

.638*** (.195)

-.074 (.195)

-.629*** (.195)

Oceania 2.295*** (.562)

-3.848*** ( .225)

1.188*** (.225)

.477** (.225)

-.535** (.225)

EU 2.336*** (.627)

-2.509*** ( .251)

-.379 (.251)

-.534** (.251)

-.316 (.251)

ROW -.500 (.422)

-.096 ( .169)

-.121 (.169)

.422** (.169)

.595*** (.222)


Table 7. Unconditional Elasticities of Derived Demand Model Elasticities

Wage Input price index

Cross-Price Exporting Country

Output Price

Own-Price

U.S. Oceaniaa EU ROWb U.S.

2.209* (1.179)c

-2.537* (1.355)

-5.684* (3.034)

-1.085*** (.029)

1.283*** (.149)

.291*** (.000)

-.165***(.052)

Oceania

5.547*** (1.358)

-6.371*** (1.660)

-14.272*** ( 3.494)

-2.229*** (.172)

1.968*** ( .034)

.442*** (.000)

.629***(.060)

EU

5.646*** (1.516)

-6.484*** (1.742)

-14.526*** ( 3.901)

-1.574*** (.000)

.415*** (.038)

1.114*** (.192)

.869***(.067)

ROW

-1.208 ( 1.021)

1.387 (1.172)

3.109 (2.626)

-.349*** (.045)

-.291*** (.025)

.070 (.129)

.395** (.000)


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